If \(X\)  and \(Y\)  are Banach Spaces, then by \(\mathcal {B}(X,Y)\)  we denote the set of all bounded (continuous) linear operators \(L:X\rightarrow Y\)  , which is itself a Banach space with the operator norm \(\Vert L\Vert =\sup _{x}\left\lbrace \Vert L(x)\Vert _Y:\Vert x\Vert =1 \right\rbrace \)  for all \(L\in \mathcal {B}(X,Y).\)  The linear operator \(L:X\rightarrow Y\)  is said to be compact if its domain is all of \(X\)  and for every bounded sequence \((x_n)\in X\) , the sequence \(\left(L(x_n)\right)\)  has a subsequence which converges in \(Y\) . The operator \(L\in \mathcal {B}(X,Y)\)  is said to be of finite rank if \(\dim R(L)<\infty \) , where \(R(L)\)  denotes the range space of \(L\) . A finite rank operator is clearly compact [1]}.
